Clarification on "有点儿" and “一点儿”

“你今天怎么样?”( Nǐ jīntiān zěnme yàng? How are you doing today?)

“一点儿累。”  ( Yìdiǎnr lèi. A bit tired.)

Unfortunately this is WRONG! It should be "有点儿累"(Yǒudiǎnr lèi ).  "But why? They both mean 'a little bit', isn't it?"  Yes, you are right.  However, there are very subtle but important differences between them.

一点儿+noun

"一点儿" is "a little bit of something, mostly substantial things ", such as 牛奶 (niúnǎi, milk),钱 (qián, money),蛋糕 (dàngāo, cake), 时间 (shíjiān, time). So it's always used before a noun and indicates the amount of the things is not very many.

有点儿 + adjective

"有点儿" is “a bit ”, it describes the degree is not a lot.  It is used before adjectives, for example 忙 (máng, busy),累 (lèi, tired),贵 (guì, expensive),难 (nán, difficult)   and often indicates a negative feeling. It can also be used before verbs related to emotions, like 喜欢 (xǐhuān),讨厌 (tǎoyàn),羡慕 (xiànmù).

有 + (一)点儿 + Noun

There is a very tricky situation, when "一点儿 + noun " is used with the verb "有"(yǒu, to have/there is),which means "somebody has a little bit amount of something" or "there is a little bit amount of something". For instance, "我有一点儿钱。"(Wǒ yǒu yìdiǎnr qián, I have a small amount of money)“冰箱里有一点牛奶。”(Bīngxiāng lǐ yǒu yìdiǎnr niúnǎi, there is a small amount of milk in the refrigerator.) What's more confusing is the "一" can be omitted! But remember it's still two words here"有 and (一)点儿"(have/there is and a little of) instead of one word “有点儿” (a little bit).

adjective+(一)点儿

Then how about "便宜一点儿"(Piányí yìdiǎnr)? “便宜”(Piányí ) is an adjective, why we are using "一点儿" instead of "有点儿"? 

Yes, you are right. But pay attention to the positions of the words. When we put "一点儿"(and the "一" can be omitted)after an adjective, it means "a bit more", which involves a comparison. “便宜一点儿” (Piányí yìdiǎnr)means “a little bit cheaper”. And "有点儿" doesn't have the same function.