I have one strongish password which I modify in a systematic and easy to remember way based on the website name. For example (and this isn't exactly what I do, obviously), say my core password is ghs78kja: on slashdot I would use as a password /DOTghs78kjaSLASH* on the New Scientist's site I would use /SCIENTISTghs78kjaNEW*.
While I understand the appeal of such a system (and tried it briefly years ago), it seems somewhat bizarre to me if you actually want any security. Yes, it will stop some random hacker who obtained a password list from site X from automatically logging into site Y by just applying the old list.
But if a hacker actually gives a crap about what he's doing and actually wants to get into your accounts, a system like this is well-known enough that he could guess your passwords to other sites once he knows one of them.
Obviously you said this isn't what you use exactly, but to really make it reasonably secure, you'd need to have a much more sophisticated method of generating password modifications for each site (e.g., disguising the name or manipulating it in a non-obvious way, performing some non-obvious modification on your "core" password based on the site name, etc.). And once you go down that road to generate something non-obvious, then you need to recreate those steps of generation every time you try to remember a password... which could be tedious and annoying unless you design it well.
Anyhow, for accounts you really don't care about, something like this sounds fine. But GP was talking about strong passwords, which should probably be more individualized for accounts you really want to keep secure.
These passwords are all unique, long, very easy to remember, and use all the character classes.
Yeah, except I'm sure they break half of the password policies at various sites anyway. That's the primary reason I started using a password manager -- even if I used a system like yours, I'd still have to remember all the random constraints on passwords for a various sites.
For example, some sites have length maximums that could be anywhere from 8 characters up. Some sites will accept a longer string when you try to login, but they won't warn you that your password must be shorter, so you keep typing in your 20-character phrase and get rejected because your password is actually the first 12 characters or whatever. And then you have sites that don't accept special characters, or sites that require special characters (but only from a certain list), or sites that don't allow you to begin your password with a number or a special character or whatever, or sites that don't accept strings of more than X letters in a row (yes, those exist, and you have to mix up the letters with numbers or special characters).... or whatever other random constraint applies.
With a password manager, I can have 30-character passwords or whatever on all the sites that accept them. If they use special characters, I can randomly generate a password with them. If they don't, I can specify a random alphanumeric password. Or whatever. And if the maximum length is 12 characters, I can specify that too without artificially limiting the length of my passwords on other sites or having to remember "Oh yeah, that site only allows a short password and it won't warn me if I try to enter my long one..." etc.
I'm not saying password managers are the best option for everything. But for remembering random website passwords, they can work pretty well.