I'm generally against too strict or complex rules to accept passwords (rules like "It must contain at least one upper-case character, one number and one symbol" and things like that, see also XKCD #936: Short complex password, or long dictionary passphrase?). Usually I prefer calculate a strengthness metric (based on special characters included and length) and if it's above a give threshold then I accept the password, no matters if it's strong because it's a 32 characters plain lower-case text or because it's 10 characters but it also contains digits in Chinese characters).
I like this approach because it's easy for end-users and, assuming well-educated users will choose passphrases, I tend to consider their passwords reasonably secure.
Users may wrongly use incredibly common sentences (my name is adriano or things like that). As soon as good statistics will be available (maybe they're even available now but I'm just not aware) an attacker who gained list of hashed passwords may use those statistics to guess passphrases.
I can't be sure about other sites security and I'd like to prevent users to reuse same password for every site they have a registration.
I thought to ask them to include a given random word in their password, something like: Type your password, be sure to include word "honey" in it. Is it reasonable? AFAIK it shouldn't weak hashing with repetitive word/pattern because word is chosen randomly from a big enough (and language specific) dictionary.
Yes they may simply append required word to existing text (for example "my name is adriano, honey") but from cracker point of view it shouldn't matter because word is random and eventually extra separators will also improve security.